L-functions
E358024
L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.
All labels observed (6)
| Label | Occurrences |
|---|---|
| L-functions canonical | 5 |
| Hecke L-functions | 3 |
| Artin L-functions | 1 |
| Artin zeta function | 1 |
| Hecke L-function | 1 |
| Rankin–Selberg L-functions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3424533 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: L-functions Context triple: [Karl Rubin, researchArea, L-functions]
-
A.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
-
B.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
-
C.
Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
-
D.
Euler products for automorphic L-functions
Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
-
E.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: L-functions Target entity description: L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.
-
A.
Dirichlet L-functions
Dirichlet L-functions are complex analytic functions built from Dirichlet characters that generalize the Riemann zeta function and play a central role in number theory, particularly in the study of primes in arithmetic progressions.
-
B.
Dedekind zeta functions
Dedekind zeta functions are number-theoretic functions attached to algebraic number fields that encode their arithmetic properties, such as the distribution of prime ideals and class numbers.
-
C.
Selberg class
The Selberg class is a collection of Dirichlet series with specific analytic properties introduced to generalize and axiomatize L-functions in number theory.
-
D.
Euler products for automorphic L-functions
Euler products for automorphic L-functions are infinite product expansions attached to automorphic representations that encode deep arithmetic information and generalize the classical Euler product of the Riemann zeta function to a broad class of L-functions in the Langlands program.
-
E.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
complex analytic function
ⓘ
mathematical object ⓘ |
| associatedWith |
algebraic varieties
ⓘ
automorphic representations ⓘ characters of Galois groups ⓘ number fields ⓘ |
| conjecturallySatisfies | generalized Riemann hypothesis ⓘ |
| definedBy | infinite series sum a_n n^{-s} ⓘ |
| encodes | arithmetic information ⓘ |
| field |
algebraic geometry
ⓘ
number theory ⓘ |
| generalizes | Riemann zeta function ⓘ |
| hasCriticalStrip | 0 < Re(s) < 1 ⓘ |
| hasDomain | complex plane ⓘ |
| hasInvariant |
conductor
ⓘ
degree ⓘ gamma factor ⓘ root number ⓘ |
| hasProperty | analytic continuation conjectured or proved beyond region of convergence ⓘ |
| hasSpecialCase |
Artin L-functions
ⓘ
surface form:
Artin L-function
Dedekind zeta functions ⓘ
surface form:
Dedekind zeta function
Dirichlet L-functions ⓘ
surface form:
Dirichlet L-function
Hasse–Weil zeta function ⓘ
surface form:
Hasse–Weil L-function
L-functions self-linksurface differs ⓘ
surface form:
Hecke L-function
L-function of a modular form ⓘ L-function of an elliptic curve ⓘ Rankin–Selberg L-function ⓘ Selberg zeta function ⓘ |
| is | meromorphic function on the complex plane ⓘ |
| oftenDefinedBy | Dirichlet series ⓘ |
| oftenHas | Euler product expansion ⓘ |
| oftenNormalizedTo | satisfy symmetric functional equation ⓘ |
| playsRoleIn |
Birch and Swinnerton-Dyer Conjecture
ⓘ
surface form:
Birch and Swinnerton-Dyer conjecture
class number formulas ⓘ modularity theorem ⓘ prime number theorem generalizations ⓘ |
| relatedTo |
Galois representations
ⓘ
automorphic forms ⓘ motives ⓘ prime numbers ⓘ |
| satisfies | functional equation relating s and 1-s ⓘ |
| studiedBy |
André Weil
ⓘ
Bernhard Riemann ⓘ Erich Hecke ⓘ Robert Langlands ⓘ |
| usedIn |
Langlands program
ⓘ
analytic number theory ⓘ arithmetic geometry ⓘ |
| zerosConjecturallyLieOn | critical line Re(s)=1/2 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: L-functions Description of subject: L-functions are complex analytic functions, often arising from number theory and algebraic geometry, that encode deep arithmetic information and generalize the Riemann zeta function.
Referenced by (12)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Artin L-functions
this entity surface form:
Artin zeta function
this entity surface form:
Hecke L-functions
this entity surface form:
Hecke L-functions
subject surface form:
Dedekind zeta function
this entity surface form:
Hecke L-functions
this entity surface form:
Rankin–Selberg L-functions
subject surface form:
L-function
this entity surface form:
Hecke L-function